11第十一章階層線性模式.DOC

11.1 11.1.1 (student-level) (personal-level) ( ) (school-level) (organization-level) ( ) 1. (disaggregation) (estimated standard errors) (type one error). (aggregation) (within-group) (1997) (hierarchical linear model, HLM) HLM (covariance components models) (Dempster, Rubin, & Tsutakawa, 1981; Longford, 1987, 1993) (multilevel linear models) (Hox, 43

44 1994; Goldstein, 1987, 1995) (mixed-effects models) (random-effects models) (Laird & Ware, 198) (random coefficient regression models) (Rosenberg, 1973) hierarchical linear models HLM HLM Lindley Smith (197)Smith (1973) (Bayesian estimation of linear models) Dempster, Laird, Rubin (1977) EM HLM Strenio, Weisberg, Bryk (1983) HLM (growth) Mason, Wong, Entwistle (1983) HLM Laird Ware (198) HLM (meta-analysis) (Draper, 1995) HLM BMDP 5V SAS MIXED (Wang, 1997)SPSS 11 MIXED GENMODHLMMLnVARCL Kreft, de Leeuw, Kim (1990) HLM MLn VARCL 1995 Journal of Educational and Behavioral Statistics HLM HLM 11.1. HLM VARCL HLM HLM

45 X Y ( (SES) (academic achievement) ) X ( X X ) Y Yi = + 1( Xi Xi) + ri, ri ~ N(0, ) ( 11-1) 0 σ 11-1 ( X X ) =0 Yˆ = 0 ) 1( X X X Y ( 1X ) S S + S S = ( 11-) Y ( X X ) YX Y X YX 1( X X ) = = = 1X S( X X ) S X + S X S X ( 1 ) L L 1 : : Y i1 Y i = = 01 0 + 11 + ( X 1 i1 ( X X. 1) + r i i1 X. ) + r i ( 11-3) 11-3 L 1 = L ( 01 = 0 11 = 1 ) Y L 1 =L 0 X X 11-1L = L 1 1 (LT ) ( L1 L ) LT

46 1 L 010 L ( 010 11 11 1 01 0 11 1 ) 1 Y L 1 L 0 X X 11-01 0 11 1 L1 L 01 11 0 1 1 1 1 ( ) LT 1 11-3 11-3 Y = 0 + 1 ( X X. ) + r, r ~ N(0, σ ) = 1,,3,... n ( 11-4) E( E( 0 1 Cov( ) ) 0 0 1,,, 1 Var( Var( ) = τ 01, 0 1 ) = τ ) = τ ρ( 11 0, 1 τ ) = 01 τ τ 11 0 11-4 HLM r γ τ γ 1 τ 11

47 τ 01 0, 1 W ( ) W 0, 1 0 1 10 01 11 W W + u + u 1 0 ( 11-5) 11-5 11-5 u 0 u 1 (random effect) γ,..., γ 11 (fixed effect) 11-5 11-4 HLM 11-6 Y W ( X X. ) W ( X X. ) + u + u ( X X. ) + 01 10 11 0 1 r ( 11-6) E( r ) = 0, Var( r ) = σ u, E u 0 1 0 u =, Var 0 u 0 1 τ = τ 10 τ τ 01 11 = T 11-6 (least square method) (independent) (normal) (constant variance) 11-6 0 + u1 ( X X. ) r u 0 u + u 1 11-6 u 0 u 1 ( X X ) 11-6 HLM LS OLS (ordinary least squares) ML (maximum likelihood) (iteration) X, W (centering) ( ) X 1. (X ). (centering around the grand mean) ( X X.. )3. (centering around the group mean) ( X X. )W 1. (W ). ( W W. ) X

48 W 0 X W ( ) 0 X W HLM ( ) X (centering around the group mean) (X - X. ) 11-4 11-5 11-6 HLM (general model) ( X ) HLM 11-7 11-8 Y = = 0 0 + X + Q q= 1 1 q + X q X + r +... + r q X q + r ~ N(0, σ ) ( 11-7) 11-7 Q+1 ( 1 Q ) Q+1 q = rq 0 + rq 1W1 + rq W +... + r S = r + r W + u q0 q s= 1 qs s q qs q W S q + u q ( 11-8) ( q = 0,1,...,Q) Q+1 W ( S = 1,..., S ) q S q +1 1 Var ( uq) = τ qq; Cov( uq, uq ) = τ qq q HLM (Bryk & Raudenbush, 199; p.) 1. ~ N(0, σ ); r. Cov ( X q, r ) = 0; 3. u = ( u 0,..., u ) ~ N( 0, T); q 4. Cov ( W s, u q ) = 0; S q

49 5. Cov( r, u q ) = 0 HLM 11.1.3 11-6 HLM (full model) (intercepts- and slopes-as- outcomes) 0 (submodels) (Bryk & Raudenbush, 199) 11.1.3.1 (one-way ANOVA with random effects) 11-4 1 = 0 Y = 0 r, r ~ N(0, σ ) ( 11-9) + 11-5 γ 0 01 = ( 11-10) 0 + u 0 11-10 11-9 Y γ + r ( 11-11) = + u0 11-11 γ u 0 ( ) r ( ) ) ( u 0

430 Var( Y ) Var( u + r = τ + σ ρ = τ /( τ + ) ρ = 0 ) σ (intraclass correlation coefficient) (cluster effect) τ ) ( τ + ) ( σ ρ ANCOVA (random-intercept model) Var Y ) τ + ( σ 11.1.3. (means-as-outcomes regression) 11-9 11-1 ( 11-1) 0 01W + u0 11-1 0 ( W ) 11-1 11-9 Y γ + r ( 11-13) = 01 W + u0 11-1 u 0 = 0 γ 01W γ u 0 (residual)u 0 ( τ ) W 0 (conditional variance) (Bryk & Raudenbush, 199, p.18) 11-10 u 0 = 0 γ (u 0 ) W

431 11.1.3.3 (one-way ANCOVA with random effects) 11-14 11-14) Y = 0 + 1 ( X X..) + r ( 11-14 ( X ) ( X X.. ) ( 11-5) γ γ 11 u 0 11-15 01 0 1 10 + u 0 ( 11-15) 11-15 ( 1 ) γ ( 1 ) 0 0 (pooled within-group regression) X. ) 10 ( 0 = Y γ 10( X. X ( X..) (adusted mean) (..) ) 11-15 11-14 Y γ + r ( 11-16) = 10( X X..) + u0 11-16 Var( ) = σ r ( X ) ANCOVA 11-16 (W ) ANCOVA 11-17 u 0 Y γ + r ( 11-17) = 01W 10( X X..) + u0

43 ANCOVA ANCOVA (classical fixed-effect ANCOVA) = H 0 : Var ) = τ 0 1 γ 10 ( 1 11 = 11.1.3.4 (random coefficients regression model) 11-4 11-18 0 1 10 + u + u 0 1 ( 11-18) u 0 τ τ01 11-18 Var = = T 11-18 u1 τ10 τ11 u 0 u 1 T (unconditional variance-covariance matrix) 11-18 γ γ 10 11-18 var( var( 0 1 ) = var( ) = var( 0 1 γ γ 10 ) = var( u ) = var( u 0 1 ) = τ ) = τ 11 ( 11-19) τ τ 11 11-18 11-4 11-0 Y ( X X. ) + r ( 11-0) 10 ( X X. ) + u0 + u1 u 0 11-0 u ( X X. ) 1 r

433 11-4 r = Y 0 1 ( X X. ) r Y 0 11-9 = ( r ) X ( ) 11.1.3.5 (a model with nonrandomly varying slopes) 11-4 u 1 0 0 1 10 01 11 W W + u 0 ( 11-1) 11-1 11-4 Y γ + r ( 11-) = 01W 10( X X. ) 11W ( X X. ) + u0 11-1 11- ( 1 ) (W ) Bryk Raudenbush (199) HLM uˆ 1 0 11.1.4

434 11-1 Y = 0 + 1 ( X X. ) + r 01W 0 10 11 1 01 10( γ11 0 1 0 + u, 1 W + u Y W X X. ) + W ( X X. ) + u + u ( X X. ) + r Y = 0 + r 0 + u 0 Y γ + u + r = 0 ρ = τ /( τ + ) σ Y = 0 + r 0 01W + u0 Y γ + W + u + r = γ 01 0 Y = 0 + 1 ( X X..) + 0 + u 0, 1 10 Y 10( X X..) + u0 + r Y = 0 + 1 ( X X. ) + r 0 + 0, 1 10 + u u 1 Y + X X. ) + u + u ( X X. ) + r γ 10 ( 0 1 r ( r r )/ r Y = 0 + 1 ( X X. ) + r 0 01 + u0, 1 10 W 11 Y γ W ( X X. ) W ( X X. + u + r = 01 10 11 ) 0 W

435 11. 11..1 HLM (1984) 0 ( ) ( T ) 40 (Bryk & Raudenbush, 199) (1994, 1995) ( ) ( ) HLM/L 5.05 HLM (Bryk & Raudenbush, Congdon, 1994) Bryk & Raudenbush (199, ) ( )

436 11.. HLM (identification number ) HLM SSM HLM ASCII ( Windows ) SASSPSSSYSTAT HLM SPSS HLM 1. SPSS ( ) IDSESACH CITY1CITY ID ID 101 ( ) 1 ( ID)31 3 ( ) 1. (aggregate) Data Aggregate Variable(s) Break Variable(s) ID Create new data file HLM_.SAV

437 3. SES_1 ( SES_AVG) SES SESACH_1 ( ACH_AVG) ACH ( ) 4. HLM File New Stat package input SSM/MDM HLM ( ) ( SASSPSS SYSTAT)

438 5. SPSS Browse HLM_1.SAV Choose Variable ( ID ) SSM 6. Browse HLM_.SAV Choose Variable ( ID ) SSM

439 7. Save Response File SSM Chen.ssm Make SSM SSM Check Stats ( ) Done 11..3 HLM (1). (). Y = 0 + r, r ~ N(0, σ ) ( 11-3) ( 11-4) 0 + u 0

440 Υ i 0 γ 40 σ ( ) τ ( u 0 ) ( ) 1. ACH Outcome Variable ( ). 3. Run Analysis Save Run the model shown 4.

441 11- γ 119.71.860 χ p u 0 306.63 39 468.347.0 r 103.974 119.71.86095% 119.711.96(.860), 119.711.96(.860) (114.115, 15.37) τ.01 (τ ) ( σ ) ρˆ ρ ˆ = τˆ /(ˆ τ + ˆ ) ( 11-5) σ =306.63/(306.63+103.974) =.304 3.04%

44 ˆ _ λ = reliability ( Y ) = τˆ /[ˆ τ + ( σˆ / n )] ( 11-6) λˆ = Σλ ˆ/ J =.914 ( 11-7) 11..4 (1). 40 40 ( ) ( ) ( ) (). 40 ( ) ) 11-4Y X X X ( ) 0 0 1 σ =Var (Y ) 11-18 γ γ 1 u 0 τ u 1 τ 11 10

443 1. SES add variable group centered ( ). SES 3.

444 11-3 t p γ 119.678.867 41.745 0.904 0.106 8.559.0 γ 10 χ p u 0 310.166 39 50.084.0 u 1 r 956.743 0.150 39 65.90.5 119.670.867 119.71 t 8.559.01 1 0.904 =[ $σ ( ANOVA)$σ ( ) ]/ $σ ( ANOVA) = (103.974956.743) /103.9746.56% τˆ 310.166, df=401=39, χ =50.084,.01 40 τ ˆ =0.150, df=39, 11 χ =65.90,.05 11..5 (1). ().

445 11-3 11-8 ( 11-8) 0 01( W W.) + u0 W W. 01 γ u 0 τ τ Var( 0 W W.) 1. SES Delete variable from model ( ). Level- Vars SES_AVG add variable grand centered ( ) 3. SES_AVG

446 4. 11-4 t p γ 119.393 1.634 73.065 γ 01.457 0.35 6.977.0 χ p u 0 81.189 38 164.906.0 r 105.466 (t=6.977, p<.01) =[ $u ( ANOVA) $u ( ) ] / $u ( ANOVA) =(306.63-81.189)/

447 306.63=0.735 73.5% ( τ = 81.189, df = 38, χ 164.906, = p<.1) ρ ˆ = τˆ /(ˆ τ + ˆ ) =9.01/(9.01+3.0)=.196 σ 11..6 (1). ( ) (). (3). 11-4 0 1 10 01 11 ( W ( W 1 1 W 1.) W.) 0 1 W W 03 13 W W 3 3 + u + u 0 1 11-9) W W. 01 γ W γ W 3 0 03 γ u 0 τ 11 γ γ 1 13 γ u 1 τ τ 11 11

448 Var( ( W 1 W 1.), W, W 3, ) 1 1. ACH Outcome Variable ( ). SES add variable group centered ( ) 3. Level- Vars SES_AVG add variable grand centered ( ) 4.

449 5. Level- Vars SES_AVG add variable grand centered ( ) 6 CITY_1 CITY_ Add variable uncentered ( ) 6. 7

450 11-5 t p γ 119.586 4.491 6.69 γ 01.86 0.506 5.581.0 γ 0-4.479 6.355-0.705.485 γ 03.635 5.041 0.469.641 γ 0.586 0.439 1.336 10 γ 11-0.014 0.03-0.61.190 γ 1 0.381 0.55 0.76.47 γ 13 0.365 0.459 0.796.431 χ p u 0 84.841 36 166.641.0 u 1 r 958.13 0.173 36 63.698.3

451 u 0 ((306.63-84.841)/ 306.63=0.733) u 0 χ u 1 u 1 ((0.15.173)/0.150=.1533) ( ) 11.3 3.04% ( 6.56%) Bryk Raudenbush (199)

45 ( 73.5%)